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MiS Preprint
28/1999
Convex integration with constraints and applications to phase transitions and partial differential equations
Stefan Müller and Vladimír Šverák
Abstract
We study solutions first order partial differential relations $Du \in K$, where $u: \Omega R^n \to R^m$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov's theory of convex integration in two ways. First, we allow for additional constraints on the minors of $Du$ and second we replace Gromov's $P$-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of `wild' solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite.
